Complete the Square
Pronunciation: /kəmˈplit ðə skwɛər/ ?
Complete the square is an
algorithm
used to convert a quadratic equation into
vertex form.^{[1]}
The complete the square algorithm is also used to derive the quadratic equation. Table
1 gives the algorithm for completing the square.
Step | Quadratic Equation | Vertex Form | Description |
1 | | Start with the equation in standard form. |
2 | | | Use the distributive property of multiplication of addition and subtraction to get x^{2}+bx inside parenthesis. |
3 | | | Simplify, if possible. |
4 | | | Add and subtract b^{2}/(4a^{2}). |
5 | | | Simplify, if possible. |
6 | | | Use the distributive property of multiplication over addition and subtraction to move -(b/2a)^{2} out of the parenthesis. |
7 | | | Simplify if possible. |
8 | | | Factor the square. |
9 | | | Subtract the constant term from both sides. The equation is now in vertex form. This demonstration will continue to derive the quadratic formula by solving for x. |
10 | | | Divide both sides by a. |
11 | | | Make 4a^{2} the common denominator. |
12 | | | Combine the fractions. |
13 | | | Take the square root of both sides. |
14 | | | Simplify. |
15 | | | Subtract b/(2a) from both sides. |
16 | | | Combine the fractions. |
17 | | | Use the symmetric property of equality to put the x on the left of the equals sign. This is the quadratic formula. |
Table 1: The complete the square algorithm |
Geometric Interpretation of Complete the Square
Step | Diagram | Description |
1 | | The square has dimensions x by x so the square represents x^{2}. The rectangle has dimensions b by x, so the rectangle represents bx. The circle represents a. |
2 | | The rectangle can be divided in half. Each half represents bx/2. |
3 | | |
4 | | Move the two rectangles around the square to form part of a larger square. |
5 | | Complete the square by adding b^{2}/4 to both sides. |
Table 2: Geometric interpretation of complete the square. |
References
- Wells, Webster. A Complete Course is Algebra for Academies and High Schools, pp 213-217. Wells' Mathematical Series. Leach, Shewell & Sandborn, 1885. (Accessed: 2010-01-16). http://www.archive.org/stream/courseincomplete00wellrich#page/213/mode/1up/search/square.
- Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 206-209. International Textbook Company, January 1963. (Accessed: 2010-01-17). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n223/mode/1up.
More Information
- McAdams, David. Vertex Form. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-12-13. http://www.allmathwords.org/article.aspx?lang=en&id=Vertex%20Form.
- Stapel, Elizabeth. Completing the Square: Finding the Vertex. PurpleMath. 2009-12-13. http://www.purplemath.com/modules/sqrvertx.htm.
Cite this article as:
Complete the Square. 2010-01-17. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/completethesquare.html.
Translations
Image Credits
Revision History
2010-01-17: Added "References" (
McAdams, David.)
2008-11-25: Changed equations to images (
McAdams, David.)
2008-10-17: Initial version (
McAdams, David.)